| L(s) = 1 | + (−0.551 + 1.30i)2-s + (−1.79 + 0.905i)3-s + (−1.39 − 1.43i)4-s + (0.750 + 1.69i)5-s + (−0.187 − 2.84i)6-s + (0.407 + 0.244i)7-s + (2.63 − 1.02i)8-s + (0.626 − 0.845i)9-s + (−2.61 + 0.0442i)10-s + (−2.70 − 2.10i)11-s + (3.80 + 1.32i)12-s + (−4.81 + 4.58i)13-s + (−0.542 + 0.396i)14-s + (−2.88 − 2.36i)15-s + (−0.122 + 3.99i)16-s + (−2.56 − 3.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.389 + 0.920i)2-s + (−1.03 + 0.522i)3-s + (−0.696 − 0.717i)4-s + (0.335 + 0.757i)5-s + (−0.0766 − 1.15i)6-s + (0.154 + 0.0923i)7-s + (0.932 − 0.361i)8-s + (0.208 − 0.281i)9-s + (−0.827 + 0.0139i)10-s + (−0.814 − 0.635i)11-s + (1.09 + 0.381i)12-s + (−1.33 + 1.27i)13-s + (−0.145 + 0.105i)14-s + (−0.743 − 0.610i)15-s + (−0.0306 + 0.999i)16-s + (−0.621 − 0.757i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00807140 - 0.00588044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00807140 - 0.00588044i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.551 - 1.30i)T \) |
| good | 3 | \( 1 + (1.79 - 0.905i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.750 - 1.69i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-0.407 - 0.244i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (2.70 + 2.10i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (4.81 - 4.58i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (2.56 + 3.12i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.447 + 2.57i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (3.45 + 1.63i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-8.03 - 4.54i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-8.24 + 1.63i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (4.64 + 2.94i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-3.25 - 3.58i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (1.08 + 3.27i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (10.4 + 5.56i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (9.40 - 5.32i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (6.27 - 6.59i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.137 + 1.86i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (4.00 + 3.45i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (2.80 + 0.415i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-7.61 + 4.56i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (0.517 + 0.156i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-8.64 + 5.47i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (5.58 - 2.63i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (4.75 + 3.17i)T + (37.1 + 89.6i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.49771148822257582763366150372, −9.997345864245466800304989961798, −8.979308512431292527114524586957, −7.915606040439261634284022039882, −6.72114325248731285258109781868, −6.34672127175717757958633998928, −4.99030474354760255236028659110, −4.68671787606235426995166944481, −2.52757897377879211823242333312, −0.00777927216561068673060820044,
1.44016287239874073390882419961, 2.81622297086647410432155629237, 4.63534015569599322521102364860, 5.19636846962827504778027443497, 6.41493759776176905921695194837, 7.77316716431806121456406572728, 8.294650738355891967067616225401, 9.708983742101178116572051798214, 10.20176287701365706916645125687, 11.06872795648168857654871105177
